By comparing the above formulation of the PD with the one in Weitz et al. [1]}, we obtain a model of degree 2 whose fixed points share the same stability properties with the one resulting from the replicator dynamics. Two interesting differences emerge from comparing the two models. The first one is that our model does not directly include the fixed points corresponding to all defectors in a depleted/replete environment, unless \(P = S\) . The second one can be found by looking at the degree of our system, which is one lower than the one by Weitz et al. This can be explained by looking at the rate of changing strategy in the pairwise difference comparison: the value obtained by the proportional rule \(\phi _{ij} = [a_j - a_i]_+\)  considers the probability of a payoff increase only, thus assigning a different weighting factor to different events. This aspect is one of the core elements in prospect theory and is the motivation that underpins our approach [2]}.
